Noise generated by cavitating single-hole and multi-hole orifices in

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Journal of Fluids and Structures 23 (2007) 163–189

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Noise generated by cavitating single-hole and multi-holeorifices in a water pipe

P. Testuda,�, P. Moussoua, A. Hirschbergb, Y. Aureganc

aLaboratoire de Mecanique des Structures Industrielles Durables, UMR CNRS-EDF 2832, 1 Avenue du General De Gaulle,

F-92141 Clamart, FrancebFluid Dynamics Laboratory, Department of Applied Physics, Technische Universiteit Eindhoven, P.O. Box 513,

5600 MB Eindhoven, The NetherlandscLaboratoire d’Acoustique de l’Universite du Maine, UMR CNRS 6613, Avenue Olivier Messiaen, F-72085 Le Mans Cedex 9, France

Received 13 December 2005; accepted 5 August 2006

Available online 27 October 2006

Abstract

This paper presents an experimental study of the acoustical effects of cavitation caused by a water flow through an

orifice. A circular-centered single-hole orifice and a multi-hole orifice are tested. Experiments are performed under

industrial conditions: the pressure drop across the orifice varies from 3 to 30 bar, corresponding to cavitation numbers

from 0.74 to 0.03. Two regimes of cavitation are discerned. In each regime, the broadband noise spectra obtained far

downstream of the orifice are presented. A nondimensional representation is proposed: in the intermediate ‘developed

cavitation’ regime, spectra collapse reasonably well; in the more intense ‘super cavitation’ regime, spectra depend

strongly on the quantity of air remaining in the water downstream of the orifice, which is revealed by the measure of the

speed of sound at the downstream transducers. In the ‘developed cavitation’ regime, whistling associated with periodic

vortex shedding is observed. The corresponding Strouhal number agrees reasonably well with literature for single-phase

flows. In the ’super cavitation’ regime, the whistling disappears.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Confined flow; Orifice; Cavitation; Broadband noise; Whistling

1. Introduction

1.1. Motivations

In industrial processes, cavitating flows are known to sometimes generate significant levels of noise and high

vibrations of structures. Some papers have been published in the last years on this topic: Au-Yang (2001), Weaver et al.

(2000), Moussou (2004).

In particular, fatigue issues have been reported recently for the configurations of a cavitating valve (Moussou et al.,

2001) and a cavitating orifice (Moussou et al., 2003). The examination of the noise generated by a cavitating device, in

e front matter r 2006 Elsevier Ltd. All rights reserved.

uidstructs.2006.08.010

ing author.

ess: [email protected] (P. Testud).

www.elsevier.com/locate/jfs

dx.doi.org/10.1016/j.jfluidstructs.2006.08.010

mailto:[email protected]

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Nomenclature

c speed of sound measured downstream of the

orifice (in m s�1)

cw speed of sound in pure water (in m s�1)

cmin minimum speed of sound (in m s�1)

d diameter of the single-hole orifice ðd ¼ 2:2�10�2 mÞ

deq single-hole equivalent diameter of the multi-

hole orifice ðdeq ¼ 2:1� 10�2 m)

dmulti diameter of the holes of the multi-hole

orifice ðdmulti ¼ 3� 10�3 mÞ

f 0 whistling frequency (in Hz)

D pipe diameter ðD ¼ 7:4� 10�2 mÞ

Sj cross section of the jet (in m2)

St Strouhal number for the whistling frequency

t orifice thickness, ðt ¼ 14� 10�3 mÞ

tp pipe wall thickness ðtp ¼ 8� 10�3 mÞ

U volume flux divided by pipe cross-sectional

area (in m s�1)

Ud volume flux divided by orifice cross-sec-

tional area (in m s�1)

Uj volume flux divided by orifice jet cross-

sectional area (in m s�1)

Gpp Power spectrum density of the pressure (in

Pa2=Hz)

Nholes number of holes for the multi-hole orifice

ðNholes ¼ 47Þ

pþ; p� forward, backward propagating plane wave

spectra (in Pa=ffiffiffiffiffiffiffiHz

p)

P1;P2 static pressure, respectively, upstream and

far downstream of the orifice

Pj static pressure at the jet (vena contracta)

Pv vapor pressure (Pvð310KÞ ¼ 5:65� 103 Pa

(Tullis, 1989))

S cross-section of the pipe (in m2)

DP static pressure difference across the orifice

(in Pa)

nwater kinematic viscosity of water [nwaterð310KÞ ¼

7:2� 10�7 m2 s�1 (Idel’cik, 1969)]

b volume fraction of gas in the water

rw density of water ðrwð310KÞ ¼ 994kgm�3)

s cavitation number

si incipient cavitation number

sch choked cavitation number

P. Testud et al. / Journal of Fluids and Structures 23 (2007) 163–189164

this study a cavitating orifice, is typically an industrial issue. It provides information which is a basis for a safer design

in terms of pipe vibrations.

1.2. Literature

In single-phase flow, an orifice generates a free jet surrounded by a dead water pressure region of uniform pressure,

cf. Fig. 1. The static pressure reaches its minimum value Pj in the jet region, also called the vena contracta, and large

eddies are generated in the shear layer separating the jet from the dead water region.

Two-phase flow transition occurs when the lowest static pressure in the fluid falls below the vapor pressure (Brennen,

1995). The level of cavitation is usually correlated with the help of a so-called cavitation number. Different definitions

exist of the cavitation number for cavitation in a flowing stream (also called hydrodynamic cavitation). They

correspond to different cavitation configurations, and are usually chosen for convenience, so that they can easily be

determined in practice:

(i)
For wake cavitation, that is cavitation round a body (i.e. an hydrofoil) or generated by a slit, the cavitation number
is commonly defined as function of the upstream conditions (Young, 1999; Brennen, 1995; Franc et al., 1999;

Lecoffre, 1994):

s ¼P0 � Pv

ð1=2ÞrLU20

, (1)

where U0 is the infinite upstream flow velocity, P0 the ambient static pressure, Pv the vapor pressure of the liquid

and rL the density of the liquid.

(ii)
For mixing cavitation, that is cavitation formed in a jet (i.e. in pumps, valves, orifices), a similar cavitation number,
as in the wake cavitation, can be used (Young, 1999; Brennen, 1995):

s ¼Pref � Pv

ð1=2ÞrLU20

, (2)

where Pref is very often defined as the downstream static pressure.

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Fig. 1. Flow through an orifice and corresponding evolution of the static pressure.

P. Testud et al. / Journal of Fluids and Structures 23 (2007) 163–189 165

We prefer to use the cavitation number, based on the pressure drop across the singularity generating the jet:

s ¼P2 � Pv

DP, (3)

where DP ¼ P1 � P2 is the pressure drop across the orifice, with P1 the static pressure upstream of the orifice and P2 the

downstream static pressure far away from the orifice. In this choice, we follow common practice in industry (Tullis,

1989; Franc et al., 1999; Lecoffre, 1994).

One should note that both those cavitation numbers lead to very similar classifications as they are related to each

other by the pressure drop coefficient of the singularity.

When the pressure Pj has a sufficiently low value, intermittent tiny cavitation bubbles are produced in the heart of the

turbulent eddies along the shear layer of the jet. This flow regime transition is called cavitation inception, and it appears

at a cavitation number of the order of 1 (when d=D ¼ 0:30) according to the data of Tullis (1989). Other references

(Numachi et al., 1960; Tullis and Govindajaran, 1973; Ball et al., 1975; Yan and Thorpe, 1990; Kugou et al., 1996; Sato

and Saito, 2001; Pan et al., 2001) are in good agreement with the values and scale effects given by Tullis (1989). Some

differences result from the influence of the variation in the dissolved gas content and in the viscosity (Keller, 1994).

As the jet pressure decreases further, more bubbles with larger radii are generated, forming a white cloud. The

pressure fluctuations increase and a characteristic shot noise can be heard. A further decrease in jet pressure induces the

formation of a large vapor pocket just downstream of the orifice, surrounding the liquid jet. The regime occurring after

this transition is called super cavitation and it exhibits the largest noise and vibration levels. In the super cavitation

regime, noise is known [see, for example, Van Wijngaarden (1972)], to be mainly generated in a shock transition

between the cavitation region and the pipe flow, at some distance downstream of the orifice. Downstream of the shock,

some residual gas (air) bubbles can persist but pure vapor bubbles have disappeared.

Cavitation indicators are used to predict the occurrence of cavitation regimes. The use of two of them has seemed

relevant, in view of our experimental results. First, a so-called incipient cavitation indicator, noted si, which predicts the

transition from a noncavitating flow to a moderately cavitating flow, that is called developed cavitation regime. Second,

a so-called choked cavitation indicator, noted sch, which predicts the transition from a moderately cavitating flow to a

super cavitating flow, with the formation and continuous presence of a vapor pocket downstream of the orifice around

the liquid jet. To calculate both those incipient and the choked cavitation indicators, scaling laws are given by Tullis

(1989). They take into account the various pressure effects and size scale effects, by means of extensive experiments on

single-hole orifices in water pipe flow. For multi-hole orifices, as mentioned in the same work, less data are available but

identical values are expected to hold.

Only a few studies provide downstream noise spectra generated by cavitating orifices (Yan et al., 1988; Bistafa et al.,

1989; Kim et al., 1997; Pan et al., 2001). A few complementary studies give the noise spectra created by cavitating valves

ARTICLE IN PRESSP. Testud et al. / Journal of Fluids and Structures 23 (2007) 163–189166

(Hassis, 1999; Martin et al., 1981). In fact, it appears that far more research has been developed on submerged water jets

(Jorgensen, 1961; Esipov and Naugol’nykh, 1975; Franklin and McMillan, 1984; Brennen, 1995; Latorre, 1997).

A comprehensive overview of the state of the art in this domain is given in Brennen (1995).

2. Experimental set-up

2.1. Tested orifices

In the piping system of French nuclear power plants, a basic configuration to obtain a pressure discharge can be

realized with a single-hole orifice. The maximum flow velocity can reach about 10m s�1 and the pressure drop 100 bar

across the orifice. This can induce high vibration levels. The orifices used are chosen in order to reduce the pipe

vibration to acceptable levels (Caillaud et al., 2006).

In our study, two orifices have been tested (see Fig. 2), as follows:

(i)
A single-hole orifice, circular, centered, with right angles and sharp edges. It has a thickness of t ¼ 14mm ðt=d ¼
0:64Þ and a diameter of d ¼ 22mm ðd=D ¼ 0:30Þ, for a pipe diameter of D ¼ 74mm. It is considered as a ‘thin’

orifice as t=dt2 (Idel’cik, 1969). In a sharp edged orifice flow, separation occurs at the upstream inlet edge. In a

thin orifice, there is no reattachment of the flow within the orifice.

(ii)
A multi-hole orifice, with Nholes ¼ 47 circular right-angled and sharp-edged perforations of diameter dmulti ¼ 3mm.
Its total open surface is practically identical to the single-hole orifice one, as it has an equivalent deq=D ¼ 0:28 ratio.This multi-hole orifice also has the same thickness of t ¼ 14mm ðt=dmulti ¼ 4:67Þ. It behaves as a thick orifice

t=d\2. The flow reattaches to the wall within the orifice.

2.2. Test rig

The test-section, as shown in Fig. 3, consists out of an open loop with a hydraulically smooth steel pipe of inner

diameter D ¼ 74mm and wall thickness tp ¼ 8mm. The orifices are placed between straight pipe sections with lengths,

respectively, equal to 42 diameters upstream and 70 diameters downstream.

The water is injected from a tank located 17m upstream from the orifice. The nitrogen pressure in the tank above the

water is controlled by a feedback system to maintain a constant main flow velocity. The water is released at atmospheric

pressure 20m downstream of the orifice. The temperature is kept equal to 310K ð�1KÞ during all experiments.

The flow velocity U is measured 26D upstream of the orifice, and the static pressures P1 and P2 are determined,

respectively, by a transducer 11D upstream and another 40D downstream of the orifice. The fluctuating pressures are

monitored by means of a combination of Kistler 701A piezo-electrical transducers and Kistler charge amplifiers. The

location of the dynamical pressure transducers is given in Fig. 4. Upstream, the transducers 1–3 are positioned, at

respectively 11D, 8D and 5D from the orifice (0.25m between consecutive transducers). Downstream, the transducers

4–10 are regularly positioned, from 7D to 39D (0.4m between consecutive transducers).

Fig. 2. Front views of the tested orifices: (a) single-hole orifice; (b) multi-hole orifice.

ARTICLE IN PRESS

Fig. 3. Scale scheme of the test rig.

Fig. 4. Location of the dynamical pressure transducers upstream and downstream of the orifice.


2.3. Experimental conditions

2.3.1. Water quality

The water used is tap water, demineralized, with pH 9 and weak conductivity. It is not degassed, hence it is expected

to be saturated with dissolved air. The volume fraction of dissolved gas (denoted by b) is high compared to other

cavitation studies. This gas content has not been measured, but an estimation, assuming saturation condition under a

temperature of T ¼ 310K or T ¼ 273K, gives for the volume fraction an order of magnitude around, respectively, 10�2

or 10�3.

It should be pointed out that the presence of dissolved gas in the water does not mean a presence of gas bubbles in the

water. Thus, the measured upstream speed of sound does not reveal any presence of gas bubbles as it is close to the one

in pure water flow.

Correcting the compressibility of the water for the influence of the elasticity of the pipe (diameter D ¼ 7:4� 10�2 m,

wall thickness e ¼ 8� 10�3 m, Young’s modulus E ¼ 2� 1011 Nm�2, Poisson ratio z ¼ 0:3), the speed of sound cth in

pure water in the pipe is given in function of the speed of sound in pure water cw (Lighthill, 1978):

cth ¼ cw 1þ rwc2wDð1� z2Þ

eE

� ��0:5

. (4)

This predicts a speed of sound of cth ¼ 1454m s�1 using cw ¼ 1523m s�1. The measured speed of sound upstream of the

orifice is 1420� 10m s�1.

2.3.2. Experimental flow conditions

Experiments are carried out at a constant flow by controlling the static pressure upstream of the orifice. The

downstream pressure P2 is imposed by the hydraulic static head of the 17.2m high vertical pipe downstream of the


orifice. Each experiment lasts about 90 s. Pressures and volume flows are provided in Table 1 for the six experiments on

the single-hole orifice, and in Table 2 for the six experiments on the multi-hole orifice.

The Reynolds number Re ¼ UD=nwater based on the pipe diameter and the water viscosity varies from 2� 105 to

5� 105; turbulence is fully developed, as usual in industrial pipes.

Higher flow regimes have been tested, but the pressure transducers downstream delivered no signal, as they were

located in a vapor pocket characteristic of the super cavitation regime. As a consequence, no acoustic data are available

in these conditions, and the corresponding hydraulic conditions are not reported in Tables 1 and 2.

2.4. Distinction of two cavitation regimes

The application of Tullis’ formulas to our experiments gives the cavitation indicators: si for the developed cavitation

and sch for the super cavitation. Compared to our observations, those cavitation regime indicators are in good

agreement:

(a)

Tab

Flow

Tab

Flow

For the single-hole orifice: siX0:93, sch ¼ 0:25. The observations, based on listening and on the measured

downstream speed of sound, indicate that all experiments (i.e. so0:74) are cavitating. Furthermore, the

downstream acoustic properties and particularly the shape of the downstream noise spectra indicate that the last

three experiments (i.e. so0:25) are in super cavitation regime.

(b)
For the multi-hole orifice: siX0:87, sch ¼ 0:20. The observations in this case indicate that all experiments (i.e.
so0:74) are cavitating. The super cavitation regime is observed for the last two experiments (i.e. so0:17).

le 1

conditions for the single-hole orifice experiments (with standard deviations)

Developed cavitation Super cavitation

U ðms�1Þ 1.91 2.38 2.90 3.75 4.08 4.42

St. deviation

ðms�1Þ

0.06 0.04 0.04 0.03 0.03 0.05

P1 ð105 PaÞ 6.3 9.2 13.3 21.4 25.0 29.5

St. deviation

ð105 PaÞ

0.3 0.3 0.6 1.4 0.7 0.8

P2 ð105 PaÞ 2.7 2.7 2.7 2.8 2.8 2.8

St. deviation

ð105 PaÞ

0.0 0.0 0.0 2.0 0.2 1.5

s 0.74 0.41 0.25 0.15 0.12 0.10

le 2

conditions for the multi-hole orifice experiments (with standard deviations)


U ðms�1Þ 2.08 2.45 2.94 3.65 4.18 4.43

St. deviation

ðms�1Þ

0.02 0.04 0.04 0.02 0.02 0.04

P1 ð105 PaÞ 6.5 6.9 12.9 19.8 26.0 28.3

St. deviation

ð105 PaÞ

0.1 0.3 0.4 1.2 0.3 0.6

P2 ð105 PaÞ 2.7 2.8 2.8 2.9 3.0 0.9

St. deviation

ð105 PaÞ

0.0 0.1 0.1 0.3 1.2 2.1

s 0.74 0.45 0.28 0.17 0.13 0.03

ARTICLE IN PRESSP. Testud et al. / Journal of Fluids and Structures 23 (2007) 163–189 169

2.5. Acoustic analysis method

In the frequency range of the study, only acoustic plane waves propagate. The issue is to determine the spectra pþ and

p� representing, respectively, the upstream and downstream traveling plane waves and for which Fourier-like analysis

holds.

From each experiment, time fluctuating-pressure signals are obtained. These data are truncated to a time interval

where the acoustic properties do not evolve significantly, i.e. on a duration of about 10 s. With the help of a reference

pressure pref , we compute the cross-spectral densities Gppref ðf Þ, which are defined as the Fourier Transform of the time

correlation (Bendat and Piersol, 1986). It is worth recalling that, for a small frequency bandwidth Df , the mean-square

value of the pressure in the frequency range ½f ; f þ Df is given by Gpp ðf ÞDf . These cross-spectra are expressed in

Pa2=Hz. In order to get a spectral expression linear with the pressure, we choose to use the following expression, in

Pa=Hz1=2 (see Appendix A for more details):

pnðf Þ ¼GpnðtÞpref ðtÞðf ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGpref ðtÞpref ðtÞðf Þ

p ð1pnp10Þ. (5)

For upstream observations, transducer 1 is chosen as reference, and for downstream observations, transducer 10 is

chosen as reference.

As plane waves propagate, the acoustic pressure at one point is the summation of the forward (in the direction of the

flow) traveling wave pþ and of the backward traveling wave p�. We assume the speed of sound to be identical in the

forward and in the backward directions, because the Mach number is low (of the order of 10�3). The identification at

each transducer of the local speed of sound and of the acoustic plane waves is carried out according to standard

intensimetry techniques (Davies et al., 1980; Boden and Abom, 1986; Hassis, 1999).

The final spectra represent an average of about 20 spectra, determined with a time signal of 10 s duration. Each

intermediate spectrum is determined with a window of 1 s duration, and the successive windows have an overlapping

ratio of 0.5 (between 0 and 1). This average is made in order to reduce the random errors.

The use of a single reference microphone may give uncertainties for determining standing waves if there is a pressure

node at a microphone. This happens when the reflection coefficient is close to unity. At low frequencies (below 500Hz),

this is the case for developed cavitation (see Fig. 6). However, as shown in Fig. 19, the microphones are not close to the

pressure nodes of the standing wave patterns. At higher frequencies, and for super cavitation, the reflection coefficient is

so low that we do not expect a problem (see Figs. 6 and 7).

2.6. Acoustic boundary conditions on both sides of the orifice

2.6.1. Acoustic boundary conditions upstream of the orifice

The acoustic conditions imposed by the test rig upstream of the orifice are quite reflecting, as illustrated in Fig. 5 by

the upstream reflection coefficient R ¼ p�=pþ, defined as the ratio between the forward and the backward propagating

plane waves. These reflecting characteristics are due to multiple partial reflections of the acoustic waves on various

elements present on the upstream part of the test rig as an open valve, a few bends and section restrictions. Those

elements have not been modified in the course of the experiments, so that these reflecting conditions not vary much

from one experiment to another.

2.6.2. Acoustic boundary conditions downstream of the orifice

The downstream acoustic boundary condition depends on the cavitation regime, developed cavitation regime or

super cavitation regime.

In developed cavitation regime (see Fig. 6), the downstream reflection coefficient has a magnitude close to 1 up to

600Hz, which indicates a strong reflecting condition. As the phase is a linear function of the frequency, the reflection

point is determined: it corresponds to the location of an open valve (at 53D downstream of the orifice). The cavity of

this valve is hence filled with a very compressible fluid, i.e. air, imposing an acoustic pressure node p0 ¼ 0.

The reflection coefficient shows values above unity in Figs. 5 and 6. This may be due to measurement inaccuracies.

Also the presence of a noise source outside the main source region is possible and would give values above unity.

In the super cavitation regime (see Fig. 7), the downstream reflection coefficient falls down below the value of 0.5. As

a first approximation, the pipe termination is then almost anechoic. This difference of behavior between the developed

cavitation regime and the super cavitation regime is not analysed in the framework of the present study.

This variation of the downstream acoustic boundary conditions is specific of the test rig. It has some impact on the

levels of the downstream spectra. This will be taken into account further in the study of the downstream noise spectra.

ARTICLE IN PRESS

10 500 1000 1500-pi

0

pi

Phas

e (r

ad)

10 500 1000 15000

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Am

plitu

de

U=2.38 m.s-1

Fig. 5. Typical acoustic reflection coefficient R ¼pþp�

upstream of the orifice (calculated at transducer 1).


It should be pointed out that, for some frequencies, the determination of the acoustical spectra is inaccurate. But

coherence values are still good enough, as illustrated in Fig 8, to allow a satisfactory fit of the spectra to determine the

speed of sound, as illustrated in Fig. 9.

3. Cavitation regimes

3.1. Hydraulic model for the pressure drop DP across the single-hole orifice

A simple model of the hydraulics of the single-hole orifice is proposed. The hydraulic conditions (pressure and Mach

number) at the jet of the orifice are estimated, hence giving some insight for a physical interpretation of the different

cavitation regimes.

This hydraulic model is a simple classical Borda-Carnot model, see, for example Durrieu et al. (2001). It assumes an

incompressible stationary single-phase flow, that is, no effect of cavitation on the hydraulics. Also, the ratio Sj=S

between the free jet cross-section and the pipe cross-section S is considered as fixed.

Firstly, mass conservation requires

SjUj ¼ SU . (6)

Secondly, the flow upstream of the jet is assumed to be an inviscid steady potential flow, so that

P1 þ12rwU2 ¼ Pj þ

12rwU2

j . (7)

Downstream of the jet, a turbulent mixing region is followed by a uniform flow of velocity U. Neglecting friction at the

walls, and assuming a thin ðt=dt2Þ orifice (with no reattachment of the flow inside the hole of the orifice), one obtains

(Sd is the cross-section of the orifice)

SPj þ rwU2j Sj ¼ SP2 þ rwU2S. (8)

The pressure Pj and the velocity Uj at the jet are deduced from Eqs. (6)–(8). The measured pressure drop, denoted by

DPmeasured, is in this case equal to the pressure drop across the orifice: DPmeasured ¼ P1 � P2. Hence, denoting by a the

contraction coefficient ða ¼ Sj=Sd Þ, this developed cavitation model gives

DPmeasured

ð1=2ÞrwU2¼

S

Sj

� �2

� 1

" #þ 2� 2

S

Sj

� �, (9)

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10 500 1000 1500-pi

0

pi

Phas

e (r

ad)

10 500 1000 15000

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Am

plitu

de

U=3.75 m.s-1

Fig. 7. Typical acoustic reflection coefficient R ¼p�pþ

downstream of the orifice (calculated at transducer 8) in super cavitation regime.

10 500 1000 1500-pi

0

pi

Phas

e (r

ad)

10 500 1000 15000

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Am

plitu

de

U=1.91 m.s-1

Fig. 6. Typical acoustic reflection coefficient R ¼p�pþ

downstream of the orifice (calculated at transducer 8) in developed cavitation

regime.


the first part representing the dissipation from upstream to the jet, and the second part the dissipation from the jet to

downstream.

Finally, after simplifications, we find

DPmeasured

ð1=2ÞrwU2¼

D

d

� �21

a� 1

" #2

. (10)

ARTICLE IN PRESS

0 500 1000 1500 2000

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

cos

(ω∆l

/c)

(p6(ω)+p8(ω)) / p7(ω)

fit with c=1425 m/s

Fig. 9. Example of the determination of the speed of sound using a fit on the spectra for three successive transducers (single-hole

orifice, U ¼ 2:38ms�1, middle transducer: 7).

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Coh

eren

ce

Fig. 8. Example of the coherence function between two successive transducers (single-hole orifice, U ¼ 2:38m s�1, between transducers

7 and 8).


In super cavitation, the downstream static sensor is in the jet region, measuring Pj . The pressure drop measured

represents the dissipation from upstream to the jet (the first part of the preceding expression):

DPmeasured

ð1=2ÞrwU2¼

1

a2D

d

� �4

� 1. (11)

ARTICLE IN PRESS

1 2 3 4 5 6 7 8180

200

220

240

260

280

300

320

U (m.s-1)

∆Pm

easu

red/

(0.

5ρw

U2 )

single-hole experiments

thin-hole developed-cavitation model: α=0.65 thin-hole super-cavitation model: α=0.65

Fig. 10. Comparison between the Borda–Carnot model with different contraction coefficients a and the single-hole orifice

measurements.

Table 3

Conditions at the jet using the Borda–Carnot model for the single-hole orifice


U ðms�1Þ 1.91 2.38 2.90 3.75 4.08 4.42

Uj ðms�1Þ 33 41 50 65 71 76

cmin ðms�1Þ 28 30 34 13 — —

Pj ð105 PaÞ 1.4 1.6 2.0 0.3 — —


Fig. 10 compares the two models using a contraction coefficient a ¼ 0:65 with experimental results (additional

experimental results in super cavitation, not shown in Table 1, are plotted), as follows:

(i)
In the developed cavitation regime ðUo3:5m s�1Þ, theory agrees qualitatively well with experiments, predicting DP
within 15%.

(ii)
In the super cavitation regime ðU43:5m s�1Þ, experimental data agree remarkably well with the model, predicting
DP within 1%. Incidentally, we observe that super cavitation does not induce a strong slope variation in the DP

versus U curve. This is mentioned in Tullis (1989) for orifices of low d=D ratio (approximately under 0.5, which is

the case here).

As a result, the model is validated as a satisfying broad estimation of the hydraulics, taking a contraction coefficient of 0.65.

This value is reasonable, as it is close to 0.61 which is the theoretical value for sharp-edged orifices with a jet from a gas to a

free space exit (Gilbarg, 1960) and it is less than 0.70, which indicates (Blevins, 1984) that the real radius of curvature of the

upstream edge of the orifice is less than 1% of the pipe diameter (this confirms that this edge is a neat sharp angle edge).

Using this model, the pressure and the velocity at the jet are estimated, see Table 3, as follows:

(a)
The pressure in the jet Pj is very close to the vapor pressure of the liquid ðPv ¼ 5:65� 103 PaÞ in super cavitation
regime. This is coherent with the stationary presence of a vapor pocket in this region. Hence, it may be an indicator

of the transition to super cavitation regime.

(b)
The velocity Uj is compared to cmin, an evaluation of the lowest speed of sound in the two-phase region of the jet.
An estimation of it can be obtained, see Van Wijngaarden (1972): the minimum speed of sound is obtained for a


vapor volume fraction in the jet of b ¼ 0:5, giving cmin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gPj=rw

p. We find (see Table 3) very low values: this is

coherent, as there is much vapor in this region. Comparing those values with the velocity of the jet flow Uj , it

appears from this crude model that the flow can be ‘supersonic’ in the jet region. This seems relevant in all

cavitation regimes, hence this supersonic transition may be characteristic of the transition between a noncavitating

and a cavitating flow: it could an indicator of incipient cavitation. Further research is needed to confirm this idea.

3.2. Hydraulic model for the pressure drop DP across the multi-hole orifice

Similarly to the single-hole orifice, a simple model of the hydraulics of the multi-hole orifice is proposed.

Contrary to the single-hole orifice, the multi-hole orifice has a ratio t=deq over 2 ðt=deq ¼ 4:7Þ, so that each orifice is

considered as thick. In that case, for noncavitating flows, there is reattachment of the flow within the orifices, followed

by a turbulent mixing region, which corresponds to a first energy loss. A second energy loss is imposed by the sudden

enlargement at the exit of the hole.

Following this description, and using similar equations as for the single-hole case, the pressure drop measured across

the orifice DPmeasured ¼ P1 � P2 for the developed cavitation regime is composed of a first part representing the

dissipation from upstream to the jet (within the orifice), and a second part representing the dissipation after the jet (Sd is

the cross-section of the orifice):

DPmeasured

ð1=2ÞrwU2¼

S

Sj

� �2

� 1

" #þ 2 1�

S

Sd

� �þ 2

S

Sd

� �2

1�Sd

Sj

� �" #. (12)

The total expression is hence the following:

DPmeasured

ð1=2ÞrwU2¼ 1� 2

D

d

� �2

þ 2D

d

� �4

1�1

aþ

1

2a2

� �. (13)

In the super cavitation regime, we assume that there is no reattachment within the orifice. The downstream static

pressure transducer is located intermediately between the two pressure losses, so that the measured pressure loss is given

by DPmeasured ¼ P1 � Pj . We should again apply Eq. (11) to determine it. As the two orifices have the same open surface,

we find the same expression as the developed cavitation model for the single-hole orifice.

Fig. 11 shows comparison of this hydraulic model with experiments, as follows:

(i)
In the developed cavitation regime, the correlation is not good for a ¼ 0:65. The pressure drop DP is
underestimated by about 30%. The thin orifice equation (Eq. (10)), however, performs better, indicating that there

is not a full reattachment within the orifice. We observe a reduction of losses for the multi-hole measurements,

which is coherent as a two-step dissipation induces less pressure losses than a single step.

(ii)
In the super cavitation regime, the correlation is again excellent, with a prediction of DP within 1%.
Conditions at the jet are estimated. We use Eq. (10) with a ¼ 0:65 as the most satisfactory for this estimation in

developed cavitation. Results are given in Table 4: in the developed cavitation regime, we get the same results as in the

single-hole case: the flow is ‘supersonic’ in the sense that the jet flow velocity is higher than the minimum speed of sound

in the region; in the super cavitation regime, the estimated pressure at the jet is at the vapor pressure. This also confirms

the result obtained for the single-hole case.

3.3. Developed cavitation visualization and temporal signal

In the developed cavitation regime, bubbles are created intermittently, as illustrated in Fig. 12.

The typical time fluctuating pressure signals obtained are difficult to distinguish from a noncavitating flow (see

Fig. 13). The fluctuations display a symmetric signal around the mean pressure, with hardly any distinguishable bursts

coming from the implosion of the bubbles. Those bursts are discussed later for super cavitation.

3.4. Whistling phenomenon in developed cavitation

Whistling is present in every developed cavitation regime of single-hole orifice experiments (it is never observed for

multi-hole orifice experiments). Evidence of whistling is particularly given on the upstream acoustical spectra, by the

sharpness of the fundamental frequency peak f 0 and the existence of several harmonics at exact multiples of f 0 (see an

ARTICLE IN PRESS

Table 4

Conditions at the jet using the Borda–Carnot model for the multi-hole orifice


U ðms�1Þ 2.08 2.45 2.94 3.65 4.18 4.43

Uj ðms�1Þ 36 43 51 63 73 77

cmin ðms�1Þ 20 — 27 — — —

Pj ð105 PaÞ 0.7 — 1.3 — —

1 2 3 4 5 6 7 8140

160

180

200

220

240

260

280

300

320

U (m.s-1)

∆Pm

easu

red/

(0.

5ρw

U2 )

multi-hole experiments

thick-hole developed-cavitation model: α=0.65

thin-hole developed-cavitation model: α=0.65

thick-hole super-cavitation model: α=0.65

single-hole experiments

Fig. 11. Evaluation of the multi-hole orifice model and comparison between the single-hole orifice measurements.


example in Fig. 14). The higher harmonics are typical of steady whistling stabilized by a nonlinear feedback effect as for

all self-sustained oscillations (Fletcher, 1979; Rockwell and Naudascher, 1979).

Scant literature has been found on whistling of cavitating orifices in pipes. A vortex shedding phenomenon in

presence of cavitation has been investigated by Sato and Saito (2001), but in the particular case of thick orifices

ðt=d\2Þ. On thin orifices, some visualizations (Moussou et al., 2003) made on other EDF experiments have shown the

possibility of the presence of vortex-shedding in cavitation regime, as can be seen in Fig. 15, from Archer et al. (2002).

The whistling frequencies are given in Table 5. They do not increase continuously with the flow velocity, as one would

expect for a hydrodynamic oscillation edge-tone like (Blake and Powell, 1983). The stable frequency is typical of an

acoustic feedback which creates and maintains the acoustic oscillation close to a resonance frequency. This is also

revealed by the locking of the phase of the time pressure signals on successive sensors, which indicates a standing wave

pattern.

Upstream of the orifice, the level of whistling is higher than downstream: hence the acoustic feedback is suspected to

happen predominantly upstream. The upstream reflection coefficient, see Fig. 5, is reflecting but irregular: no clear

acoustic reflection point can be identified (as it can be for the cavity of a valve downstream, see Section 2.6.2). This

irregular shape is due to the intricacy of the upstream rig design, with a succession of elbows, slight restrictions of

sections and some open valves.

It is also worth mentioning that the whistling frequency does not coincide with any downstream natural acoustic

frequency. This is coherent with the acoustic uncoupling observed from both sides of the orifice.

The single-hole orifice has a ratio t=d ¼ 0:6, inferior to 2, so that it is considered as a thin orifice. We assume that the

whistling phenomenon is influenced by the thickness of the orifice t, rather than its diameter. It is also natural to use the

ARTICLE IN PRESS

Fig. 12. From other experiments at EDF (Archer et al., 2002), visualization of the developed cavitation regime for a single-hole orifice

(d=D ¼ 0:30, t=d ¼ 0:10, D ¼ 2:66� 10�1 m) with s ¼ 0:49 and U ¼ 1:50m s�1 (DP ¼ 3:1� 105 Pa, P1 ¼ 4:6� 105 Pa). White bubble

clouds are observed around the jet formed at the orifice.

40 40.05 40.1 40.15-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time (s)

Dyn

amic

al p

ress

ure

sign

al a

t sen

sor

9 (1

05 Pa)

σ = 0.41

Fig. 13. Typical dynamical time pressure signal from a downstream transducer for the developed cavitation regime (here for the single-

hole orifice at s ¼ 0:41).


velocity at the orifice Ud as a relevant scaling velocity. Hence the Strouhal number is defined as

St ¼f 0t

Ud

, (14)

and values are reported in Table 5.

ARTICLE IN PRESS

0 200 400 600 800 1000

104

105

106

107

108

109

Frequency (Hz)

p−2 (

Pa2 /H

z)

U=1.91 m.s-1

f0

2 f0

Fig. 14. Detection of whistling frequency f 0 with harmonics on upstream plane wave spectra p� (here for the single-hole orifice at

U ¼ 1:91ms�1, f 0 ¼ 359Hz and the first harmonics at 718Hz).

Fig. 15. From other experiments at EDF (Archer et al., 2002), visualization of a cavitating and whistling single-hole orifice

(d=D ¼ 0:30, t=d ¼ 0:10, D ¼ 2:6� 10�1 m) with s ¼ 0:35 and U ¼ 1:97m s�1 (DP ¼ 5:3� 105 Pa, P1 ¼ 7:2� 105 Pa).


The Strouhal number is obtained in the range 0.18–0.26. These values are close to data on orifices: Anderson (1953)

finds a Strouhal number around 0.2 for whistling orifices in air with a free air exit. In Anderson’s experiments, the

orifices were a bit thinner (with 0:2pt=Dp0:5, whereas here t=D ¼ 0:19) and the Reynolds numbers smaller

(Udt=nairH103 with nair the kinematic viscosity of air, whereas here Ud t=nwaterH105).

The multi-hole orifice is a thick one, as t=dmulti ¼ 4:7. Hence it is no surprise that it does not whistle. The flow

reattaches itself inside each hole, which stabilizes the shear layers of the jet and hence prevents whistling in the same

manner as observed for the single-hole orifice.

ARTICLE IN PRESS

Table 5

Strouhal number St ¼ f 0t=Ud of the whistling frequency f 0 (only observed in the single-hole orifice case)

U ðms�1Þ 1.91 1.91 2.38 2.38 2.90 2.90

c ðms�1Þ 1390 1200 660 1420 1130 1420

f 0 (Hz) 359 397 421 436 427 434

St 0.23 0.26 0.22 0.23 0.18 0.19

Fig. 16. From other experiments at EDF, visualization of the super cavitation regime for a single-hole orifice.


3.5. Super cavitation visualization and typical temporal signal

In the super cavitation regime, a vapor pocket is created in the jet region, as illustrated in Fig. 16.

When the vapor pocket expands and reaches the downstream transducers, those transducers no longer deliver any

acoustical signal. This constitues an evidence for the existence of the vapor pocket.

It appears that the length of the vapor pocket increases quickly with flow velocity. The length of the vapor pocket

increases from about 7D at U0 ¼ 3:8m s�1 to 18D at U0 ¼ 4:4m s�1 and is larger than 38D at U0 ¼ 4:7m s�1 for the

single-hole orifice. For the multi-hole orifice, the length of the vapor pocket increases from around 7D at U0 ¼ 4:2m s�1

to larger than 38D at U0 ¼ 4:5m s�1.

The typical time-fluctuating pressure signals obtained are very different from those in the developed cavitation regime

(see Fig. 17). They are asymmetric around the mean pressure, exhibiting very large positive spikes, up to 20 bar, linked

to the collapse of bubbles.

Focusing on time signals (see an illustration in Fig. 18), it is seen that the phenomenon of bubble implosion is

characterized first by a decrease in negative values of the fluctuating pressure with a characteristic duration of a tens of

milliseconds, followed by an abrupt increase of the pressure, a ‘spike’, with a characteristic duration of 1ms. By

modeling this bubble volume evolution with a monopole source, which is a common and satisfactory simple model; see,

for instance, Brennen (1995), the first decrease is linked with a decrease of the bubble size from its initial size to a value

close to zero; the second part of the signal is linked with a pressure shock wave originating from the abrupt collision of

the water particles. However, extremely high amplitudes are observed in comparison to Brennen (1995) (cf. Fig 3.19).

This can be due either to the large size of our bubbles or to the effect of confinement.

ARTICLE IN PRESS

40 40.05 40.1 40.15-5

0

5

10

15

20

Time (s)

Dyn

amic

al p

ress

ure

sign

al a

t sen

sor

9 (1

05 Pa)

σ = 0.10

Fig. 17. Spurious pressure pulses (spikes) from collapsing bubbles on dynamical time pressure signal from a downstream transducer in

super cavitation regime (here for the single-hole orifice at s ¼ 0:10).

47.07 47.075 47.08 47.085 47.09 47.095 47.1-5

0

5

10

15

20

Time (s)

Fluc

tuat

ing

pres

sure

(x1

05 Pa)

Fig. 18. Cavitation peak in super cavitation regime (single-hole orifice at s ¼ 0:10).


4. Results in the developed cavitation regime

The single-hole and the multi-hole orifice developed cavitation regimes are presented together, as they show similar

acoustical behavior. Firstly, results on the acoustic features of this cavitation regime are introduced: the observed

variations of the downstream speed of sound and the presence of resonance frequencies in downstream spectra. Then

the downstream spectra, depending on those acoustical features, are presented and discussed.


4.1. Acoustic features

4.1.1. Spontaneous variations of the downstream speed of sound

For a fixed set of hydraulic conditions in the developed cavitation regime, the downstream speed of sound may

spontaneously evolve. For instance, the single-hole orifice experiment at U ¼ 2:38m s�1 shows the downstream speed of

sound evolving from 660m s�1 (during the first 20 s) to 1420m s�1 (till the end of the experiment: 90 s).

This variation of the speed of sound indicates that air bubbles are present, in varying quantity, in the water far

downstream of the orifice. For instance, a value of 660m s�1 indicates a volume gas fraction in the water between 10�3

and 10�4, assuming the static pressure being between Pj and P2 [we use a classic formula; see, for example, Van

Wijngaarden (1972)]. On the contrary, a value of 1420m s�1 indicates a negligible content of air in the water, as it is

close to the speed of sound in pure water [which equals to 1454m s�1, as presented in Section 2.3.1].

Cavitation bubbles, when they are formed, are originally mainly constituted of vapor. During their lifetime, they are

gradually filled with air, due to the diffusion of the dissolved gas present in the water surrounding them. As they drift

downstream, moving away from their region of creation, they reach regions where the pressure recovers higher levels.

Pure vapor bubbles cannot persist, those bubbles remaining far downstream of the orifice are mainly filled with air (see

for example Fig. 12). Consequently, the observed variation of the quantity of air bubbles is suspected to be due to an

inhomogeneity of the dissolved gas content in the injected water. This inhomogeneity may be related to temperature

variations in the experimental installations. This hypothesis is all the more plausible as the water used has not received

any degassing treatment, hence having a fluctuating and high dissolved gas content, not measured but estimated around

10�2 or 10�3 (values at saturation conditions for T ¼ 273 and 310K, respectively). In some experiments, the change in

residual air bubble content occurs after the water from the pipe segment between the orifice and the tank has been

evacuated and ‘fresh’ tank water has started to flow through the orifice.

These variations of the downstream speed of sound during each experiment have some influence on the acoustical

behavior downstream of the orifice: the values of the natural frequencies, appearing downstream, are altered

proportionally with the speed of sound.

The propagating waves are subjected to two-phase flow damping, as already mentioned by Hassis (1999). As regards this

last effect, no significant variation of the propagating wave amplitude could be measured along the downstream sensors,

but the downstream acoustical reflection coefficient appears to vary significantly with the downstream speed of sound.

4.1.2. Acoustical uncoupling from both sides of the orifice

An acoustical uncoupling is observed between acoustical spectra upstream and downstream of the orifice: the natural

frequencies present downstream are strongly attenuated on upstream spectra; the whistling, when present, is visible on

upstream spectra, but hardly on downstream spectra; and, furthermore, the background noise on downstream spectra is higher

than the one on upstream spectra, approximately from a factor 2 (for U ¼ 1:91m s�1) up to 7 (for U ¼ 2:90m s�1) for the

single-hole orifice, and much more significantly for the multi-hole orifice, with an approximately constant factor of about 10.

This acoustical uncoupling is an effect of cavitation as there is choking (indicated in Table 3).

4.1.3. Presence of natural modes downstream

In this developed cavitation regime, resonance frequencies are systematically observed in acoustical spectra

downstream of the orifice (both for the single-hole and the multi-hole).

The acoustic boundary conditions are of a similar type, as the frequencies are of the form: f n ¼ nf 1, f 1 being the first

resonance frequency. More precisely, these acoustic boundary conditions can be identified by extrapolating the standing

wave patterns at those resonance frequencies. This extrapolation is made possible as a series of transducers (7 in

number) is present downstream of the orifice. As a result, we find two acoustic pressure nodes p0 ¼ 0 (see Fig. 19),

discussed above, as follows:

(i)
One acoustic pressure node is found far downstream of the orifice, at 52D ð�1DÞ. It is the result of the influence of a
cavity of an open valve filled with air. The reflection coefficient imposed by such a cavity has been presented in

Fig. 6. The magnitude of the reflection coefficient jRj is close to 1, which confirms the acoustic influence of this cavity.

(ii)
Another is found just downstream of the orifice, at 4D ð�1DÞ. It is likely to be caused by a cavitation cloud.
4.2. Noise spectra generated downstream

Noise spectra generated in the pipe downstream of the orifice for the developed cavitation regime are presented in this

section.

ARTICLE IN PRESS

-1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5x 104

Distance (m)

Rea

l par

t of

the

cros

s-sp

ectr

a (P

a/H

z0.5 )

↑ Orifice ↑ Downstream valve

p’~0p’~0

Fig. 19. In the developed cavitation regime, two acoustic boundary conditions p0 ’ 0 are found downstream of the orifice, hence

natural frequencies appear, pointed out by the interpolation of the values of the cross-spectra at those first natural frequencies (here for

the single-hole orifice at U ¼ 2:38m s�1, c ¼ 1420m s�1 and five natural frequencies: � 188Hz, m 386Hz, . 574Hz, ’ 770Hz,

E 957Hz).


The downstream transducers give a far-field measurement, as they are not located in the source region mainly

constituted of bubble implosions located just downstream of the orifice. The acoustical power measured at downstream

transducers represents the noise generated in the pipe. Under the assumptions of plane-wave propagation and no

influence of the flow (the Mach number is around 10�3), the acoustical power takes the expression: Sðpþ2 � p�2Þ=ðrwcwÞ

(Morfey, 1971).

It is furthermore assumed that the backward propagating wave spectrum of p� is small enough to be neglected. Even

if acoustical reflection is observed to occur downstream, this estimation of the source of noise is considered satisfactory

enough as an order of magnitude (this is all the more relevant as logarithmic representation is used). Thus, it comes to

the source power being represented by the quantity Spþ2=ðrwcwÞ.

Of course, this representation is no more valid for the discrete frequencies which are whistling harmonics and

resonance frequencies, as the backward propagating wave has great influence in those cases.

The acoustical power spectra obtained downstream of the orifice for the developed cavitation regime are given in

Fig. 20 for the single-hole orifice and in Fig. 21 for the multi-hole orifice. The rejection frequency corresponding to a

wavelength of half of the distance between two successive transducers and depending on the measured speed of sound

has been excluded from those results. The following observations are made on those spectra:

(a)
spectra exhibit a hump form in the upper frequency range (above 200Hz); this hump form is typical of cavitational
noise (Martin et al., 1981; Brennen, 1995);

(b)
the single-hole orifice generates remarkably much more noise (by a factor of about 102 in the acoustical power) than
the multi-hole orifice.

4.3. Nondimensional analysis and representation

4.3.1. Choice of the scaling variables for the noise spectra

Scaling is proposed in order to obtain a dimensionless representation of the acoustical source power.

Cavitation noise from the implosions of bubbles is assumed to be predominant in order to scale the noise spectra.

Hence, broadband turbulence noise from the mixing region downstream, whistling and resonance frequencies are not

ARTICLE IN PRESS

101 102 10310-6

10-5

10-4

10-3

10-2

10-1

100

f (in Hz)

S p+

2 /(ρ w

c) (

in W

/Hz)

σ=0.74, c=1280 m/s

σ=0.41, c=660 m/s

σ=0.25, c=1430 m/s

Fig. 20. Acoustical power spectra in developed cavitation (single-hole orifice).

101 102 10310-7

10-6

10-5

10-4

10-3

10-2

10-1

f (in Hz)

p+2 S

/(ρ

c) (

in W

/Hz)

σ=0.74, c=1420 m/s

σ=0.45, c=1420 m/s

σ=0.28, c=1425 m/s

σ=0.17, c=1425 m/s

Fig. 21. Acoustical power spectra in developed cavitation (multi-hole orifice).


taken into account to choose the scaling variables. This assumption of predominance of cavitation noise is globally

valid, but seems to fail at low frequencies (in this work, below 200–300Hz approximately). Also, it is assumed that

whistling does not alter cavitation noise, generalizing the hypothesis that broadband noise is not affected by whistling

[as shown in Verge (1995) for a flue organ pipe].

Following Blake (1986), the amplitude of noise produced by cavitation should be made dimensionless by dividing

with the downstream pressure, and not the pressure drop, when using a Rayleigh–Plesset bubble dynamic model for a

spherical isolated free bubble. However, ring vortices generated by an orifice are not isolated bubbles in free space, so

ARTICLE IN PRESS

Fig. 22. Choice of the variables d and Ud for the scaling of the noise spectra in the developed cavitation regime for the single-hole

orifice.


that this scaling should not hold in the case of the present study, as it also does not for sheet cavitation on airfoils

(Keller, 1994).

As we lack a precise model, the scaling data are chosen for the sake of simplicity: the basic idea, as shown in Fig. 22,

relies on the fact that, in the developed cavitation regime, the bubbles are created in the mixing high-shear region of the

jet.

For the single-hole orifice experiments, the velocity Ud and the orifice diameter d are representative of the conditions

in this region, hence those quantities are used in order to scale the noise spectra in this regime. Furthermore, the scaling

pressure is defined as the pressure drop DP across the orifice, which is a measure of the kinetic energy density in the jet.

Hence for the single-hole orifice in developed cavitation, fd=Ud is the nondimensional frequency and pþ2 Ud=ðDP2dÞ is

the nondimensional magnitude.

For the multi-hole orifice experiments, we assume the noise issuing from incoherent Nholes sources.

Each source represents the radiation of one hole. It radiates on a characteristic surface of S=Nholes. The strength of

each source is assumed to be independent of the environment of the source. This assumption is natural, as we have

previously supposed (see previous section) that the noise generated by the single-hole orifice does not depend on the

diameter of the pipe. However, it should be pointed out that this assumption is wrong when whistling occurs.

In this model, the total acoustic power P measured downstream is a summation of the acoustic power Peach source

emitted by each source [the key element is that sources are supposed to be incoherent between each other, see Pierce

(1981)]:

P ¼ NholesPeach source. (15)

The acoustic power of each source Peach source is, by definition, the total acoustic intensity flux I multiplied the surface of

this source:

Peach source ¼ IS=Nholes. (16)

Hence the total acoustic power is

P ¼ SI . (17)

The total acoustic power is consequently independent of the number of holes. As previously, we ignore any downstream

reflections, so that I ¼ pþ2=ðrcÞ.

This argumentation based on energy considerations can also be conducted in terms of forces: if the source is

represented as a force acting on the orifice, taking the form Spþ, the total force imposed on the multi-hole orifice is due

to the contribution of the forces imposed by Nholes equivalent single-hole orifices with open surface S=Nholes. Those

forces are supposed to be uncorrelated with each other. Thus, the total force squared equals Nholes times the force

squared due to one hole. As the force squared of one hole is ðpþS=NholesÞ2, the total force squared is expressed as:

NholesðpþS=NholesÞ

2, or by simplifying: ðpþSÞ2=Nholes.

The scaling of the acoustic power is based on each source of surface S=Nholes. The scaling velocity is the velocity at

the orifice, which is taken equal to the velocity of the single-hole orifice Ud , as the open surface of the two orifices are

very similar, and the scaling length is the diameter of one hole dmulti. In conclusion for the multi-hole orifice in

developed cavitation: fdmulti=Ud is the nondimensional frequency, and pþ2 Ud=ðDP2dmultiÞ is the nondimensional

magnitude.

4.3.2. Nondimensional noise spectra generated downstream

The dimensionless acoustical power spectra obtained downstream of the orifice for the developed cavitation regime

are given in Fig. 23 for the single-hole orifice and in Fig. 24 for the multi-hole orifice. The following observations are

worth noting.

ARTICLE IN PRESS

10-1 10010-6

10-5

10-4

10-3

10-2

10-1

100

f d/Ud (10<f<2000 Hz)

p+2 U

d / (

∆P2 d

)

σ=0.74, c=1280 m/s

σ=0.41, c=660 m/s

σ=0.25, c=1430 m/s

Fig. 23. Nondimensional acoustical power spectra in developed cavitation (single-hole orifice). Straight line: dimensionless turbulence

level estimation of Moussou (2005).

10-2 10-1

10-6

10-4

10-2

f dmulti/Ud (10<f<2000 Hz)

p+2 U

d / (

∆P2 d

mul

ti)

σ=0.74, c=1430 m/s

σ=0.45, c=1430 m/s

σ=0.28, c=1430 m/s

σ=0.17, c=1435 m/s

Fig. 24. Non-dimensional acoustical power spectra in developed cavitation (multi-hole orifice).


(a)
The scaling for the single-hole orifice and the multi-hole orifice is efficient as the nondimensional noise spectra
collapse for each orifice. This is illustrated in Fig. 23 for the single-hole orifice and Fig. 24 for the multi-hole orifice.

Hence it is found that these spectra do not depend significantly on the cavitation number or on the downstream

speed of sound. However, the dispersion of the scaling variables is weak, so that additional data should be added to

confirm this result.


(b)
The different scaling used for the single-hole and the multi-hole orifice is rather efficient as it makes the levels of the
two types of orifices closer to each other. However, the nondimensional level of noise of the single-hole orifice is still

higher, with a ratio of 10, than the one of the multi-hole orifice experiments; see Figs. 23 and 24.

(c)
We compare the cavitation noise with a standard turbulence noise from a noncavitating orifice in the low frequency
range. Indeed, in this range of frequency, the level of noise is expected to be mainly determined by turbulence noise

(Martin et al., 1981). We use a nondimensional turbulence noise level proposed by Moussou (2005). In this model,

the scaling is based on empirical data obtained with simple singularities (single-hole orifice, valve) in water-pipe

flow: the level of noise is assumed to depend only on a Strouhal number based on the pipe diameter and the

pipe flow velocity. We apply this model with the values of the pipe diameter D and a pipe flow velocity of 2:20m s�1

and compare it to the single-hole orifice nondimensional noise, see Fig. 23. As expected, the turbulence level

fits rather well the cavitation noise at low frequencies, with a good estimation of the slope; the cavitation noise is

much stronger than the turbulence noise above the low-frequency range, which is a well-known result Brennen

(1995).

5. Results in the super cavitation regime

5.1. Acoustic features

The super cavitation experiments exhibit two main differences compared to the developed cavitation experiments, as

follows:

(i)
The downstream speed of sound appears to be quite constant during each experiment.
(ii)
No resonance frequencies are found on downstream spectra. The downstream reflection coefficient is much lower
than in the developed cavitation case (as previously shown in Fig. 7). In this case, the cavity of the downstream

valve does not reflect the acoustic waves.

As for the developed cavitation case, strong acoustic uncoupling is observed from both sides of the orifice. The effect

is much more obvious here, with an average ratio between downstream and upstream power spectra of about 2–30 for

the single-hole orifice case, and about 10–50 for the multi-hole orifice case.

5.2. Noise spectra generated downstream

Noise spectra obtained downstream are given in Fig. 25 for the single-hole orifice and in Fig. 26 for the multi-hole

orifice (as previously, the rejection frequencies being excluded). We note the following points:

(i)
The typical cavitational hump form is observed, as in the developed cavitation case, but with much more evidence.
The level of the hump is higher than for the developed cavitation regime; also, the frequency peak is smaller: those

tendencies when the cavitation number increases confirm literature data Brennen (1995).

(ii)
Also, as in the developed cavitation case, the single-hole orifice is clearly more noisy (with an approximate factor of
10 on acoustical power spectra) than the multi-hole orifice.

An important result is illustrated in Figs. 26 and 27: the part of the spectra succeeding the hump peak frequency

depend strongly on the downstream speed of sound. Two-phase flow attenuations phenomena are supposed to be the

cause of this observation.

6. Conclusion

A single-hole and a multi-hole cavitating orifice in a water pipe have been studied experimentally under industrial

conditions, i.e. with a pressure drop varying from 3 to 30 bar and a cavitation number in the range 0:03psp0:74.In the regime of developed cavitation, whistling is observed for the single-hole orifice. This occurs at a Strouhal

number based on the orifice thickness with a value close to 0.2.

Our results indicate that in the developed cavitation regime, a multi-hole orifice is much more silent than a single-hole

orifice with the same total cross-sectional opening. This might be partially explained by the absence of correlation


between the sound produced by different holes in the absence of whistling. No such difference is found in super

cavitation. This is explained by the fact that, in the super cavitation regime, sound is produced far downstream of the

orifice rather than in a near region downstream of the orifice.

A nondimensional noise source is proposed for the developed cavitation regime (cf. Section 4.3). The source model

seems satisfactory for the developed cavitation case.

The results presented are certainly limited and call for further research. In particular, it could be interesting to vary

the shape of the orifices and repeat the experiments with degassed water.

101 102 103

10-4

10-3

10-2

10-1

100

101

f (in Hz)

p+2 S

/ (ρ

c) (

in W

/Hz)

σ=0.15, c=280 m/s

σ=0.15, c=1420 m/s

σ=0.12, c=120 m/s

σ=0.12, c=1310 m/s

σ=0.10, c=1230 m/s

σ=0.10, c=340 m/s

Fig. 25. Acoustical power spectra in super cavitation (single-hole orifice).

101 102 103

10-4

10-3

10-2

10-1

100

101

f (in Hz)

p+2 S

/ (ρ

c) (

in W

/Hz)

σ=0.03, c=1400 m/s

σ=0.13, c=1010 m/s

σ=0.13, c=700 m/s

Fig. 26. Acoustical power spectra in super cavitation (multi-hole orifice).

ARTICLE IN PRESS

101 102 103

10-4

10-3

10-2

10-1

100

101

f (in Hz)

p+2 S

/ (ρ

c) (

in W

/Hz)

σ=0.15, c=1420 m/s

σ=0.15, c=1300 m/s

σ=0.15, c=610 m/s

σ=0.15, c=280 m/s

Fig. 27. Super cavitation (single-hole orifice): influence of the speed of sound on the noise spectra.


Acknowledgment

The experiments have been designed and realized by MM. S. Caillaud, Ch. Martin and L. Paulhiac.

Appendix A. Acoustic analysis

We assume that only plane waves propagate in the pipe. First, cross-spectral density functions Gppref are used. They are

well suited to acoustic analysis, because they eliminate nonpropagating noise present in time measurements. Indeed, if we

decompose time pressure measurements pðtÞ into a nonpropagating signal pnonpropðtÞ and a propagating one, we have

pðtÞ ¼ pþðtÞ þ p�ðtÞ þ pnonpropðtÞ. (A.1)

Having defined and fixed a reference sensor and applying the cross-spectral density functions, we get an expression where the

nonpropagating pressure is removed because it is not correlated to the propagating pressure:

GpðtÞpref ðtÞðoÞ ¼ GpþðtÞpref ðtÞðoÞ þ Gp�ðtÞpref ðtÞðoÞ. (A.2)

Second, we define quantities Gppref =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGprefpref

pwhich can be handled like Fourier Transforms, and which are almost

independent of the reference pressure. Indeed, if we assume a perfect coherence between the point of observation and the

reference sensor, we have

jGppref j2

Gpp Gprefpref

¼ 1. (A.3)

In this case the quantity Gppref =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGprefpref

pdoes not depend on the reference sensor. In practice, the reference sensor is taken

closest to the points of observation.

Finally, we calculate plane waves. Considering two consecutive sensors, denoted n and n þ 1, plane waves

propagation and the linearity characteristics of the cross-spectral density function give

Gpþnþ1

prefðoÞ ¼ Gpþn pref

ðoÞe�jot, (A.4)

Gp�nþ1

pref ðoÞ ¼ Gp�n pref ðoÞeþjot, (A.5)


with t being the time of flight of the wave between the two sensors. For the sake of ease, we note pn for Gpnpref =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiGprefpref

p.

By means of Eqs. (A.2), (A.4) and (A.5), we get the plane wave spectra:

pþn ðoÞ ¼

pnðoÞ eþjot þ pnþ1ðoÞ2j sinðotÞ

, (A.6)

p�n ðoÞ ¼

pnðoÞ e�jot þ pnþ1ðoÞ

�2j sinðotÞ, (A.7)

for upstream measurements ð1pnp2Þ and downstream measurements ð4pnp9Þ. These spectra are expressed in

Pa=ffiffiffiffiffiffiffiHz

p.

The reflection coefficient is defined as p � =pþ downstream of the orifice, and by p þ =p� upstream, because we

assume the source of sound to be located in the region of the orifice. For example, if a zero pressure condition is located

far downstream of the orifice, and if T is the time of flight toward this pressure node, the reflection coefficient at this

current point is �e�2joT .

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